# Courses tagged with "Taking derivatives" (26)

In ASTR101, you will be introduced to our current understanding of the universe and how we have come to this understanding. We will start with the ancient Greeks and their belief that the universe was an orderly place capable of being understood. We will continue through history, as we acquired more information on the nature of the universe and our models of the universe changed to reflect this. This will take us through several different worldviews. As noted above, we will begin with the Greek worldview, which was characterized by the belief that the earth was the immovable center of the universe; this was known as the “geocentric” model. Although this worldview is wrong in many of its details, it was a very important first step. It explained the universe well enough that it lasted almost two thousand years. By 1600, this belief was beginning to be challenged by such people as Copernicus, Kepler, and Galileo; finally, it was completely done away with by the physics of Newton. By 1700, the…

This course is designed to introduce you to the study of Calculus. You will learn concrete applications of how calculus is used and, more importantly, why it works. Calculus is not a new discipline; it has been around since the days of Archimedes. However, Isaac Newton and Gottfried Leibniz, two 17th-century European mathematicians concurrently working on the same intellectual discovery hundreds of miles apart, were responsible for developing the field as we know it today. This brings us to our first question, what is today's Calculus? In its simplest terms, calculus is the study of functions, rates of change, and continuity. While you may have cultivated a basic understanding of functions in previous math courses, in this course you will come to a more advanced understanding of their complexity, learning to take a closer look at their behaviors and nuances. In this course, we will address three major topics: limits, derivatives, and integrals, as well as study their respective foundations and a…

This course is the second installment of Single-Variable Calculus. In Part I (MA101 [1]), we studied limits, derivatives, and basic integrals as a means to understand the behavior of functions. While this end goal remains the same, we will now focus on adapting what we have learned to applications. By the end of this course, you should have a solid understanding of functions and how they behave. You should also be able to apply the concepts we have learned in both Parts I and II of Single-Variable Calculus to a variety of situations. We will begin by revisiting and building upon what we know about the integral. We will then explore the mathematical applications of integration before delving into the second major topic of this course: series. The course will conclude with an introduction to differential equations. [1] http:///courses/ma101/…

Differential equations are, in addition to a topic of study in mathematics, the main language in which the laws and phenomena of science are expressed. In its most basic sense, a differential equation is an expression that describes how a system changes from one moment of time to another, or from one point in space to another. When working with differential equations, the ultimate goal is to move from a microscopic view of relevant physics to a macroscopic view of the behavior of a system as a whole. Let’s look at a simple differential equation. From previous math and physics courses, we know that a car that is constantly accelerating in the x-direction, for example, obeys the equation d2x/dt2 = a, where a is the applied acceleration. This equation has two derivations with respect to time, so it is a second-order differential equation; because it has derivations with respect to only one variable (in this example, time), it is known as an ordinary differential equation, or an ODE. Let’s say t…

This survey chemistry course is designed to introduce students to the world of chemistry. Chemistry was born in ancient Egypt, when the principles of chemistry were first identified, studied, and applied in order to extract metal from ores, make alcoholic beverages, glaze pottery, turn fat into soap, and much more. What began as a quest to build better weapons or create potions capable of ensuring everlasting life has since become the foundation of modern science. Take a look around you: chemistry makes up almost everything you touch, see, and feel, from the shampoo you used this morning to the plastic container that holds your lunch. In this course, we will study chemistry from the ground up, learning the basics of the atom and its behavior. We will apply this knowledge to understand the chemical properties of matter and the changes and reactions that take place in all types of matter.

This course will survey physics concepts and their respective applications. It is intended as a basic introduction to the current physical understanding of our universe. Originally part of “Natural Philosophy,” the first scientific studies were conducted after Thales of Miletus established a rational basis for the understanding of natural phenomena circa 600 BCE. One of the Seven Sages of Greek philosophy, Thales sought to identify the substances that make up the natural world and explain how they produce the physical phenomena we observe. Prior to Thales, humans had explained events by attributing supernatural causes to them; his work represents the very beginning of scientific analysis. The Scientific Method used today builds on this early foundation, but adds the essential underpinnings of evidence based on experiments or observation. Briefly, the modern scientific method involves forming a hypothesis about the cause of a general phenomenon, using that hypothetical model to predict the outc…

The physics of the Universe appears to be dominated by the effects of four fundamental forces: gravity, electromagnetism, and weak and strong nuclear forces. These control how matter, energy, space, and time interact to produce our physical world. All other forces, such as the force you exert in standing up, are ultimately derived from these fundamental forces. We have direct daily experience with two of these forces: gravity and electromagnetism. Consider, for example, the everyday sight of a person sitting on a chair. The force holding the person on the chair is gravitational, while that gravitational force is balanced by material forces that “push up” to keep the individual in place, and these forces are the direct result of electromagnetic forces on the nanoscale. On a larger stage, gravity holds the celestial bodies in their orbits, while we see the Universe by the electromagnetic radiation (light, for example) with which it is filled. The electromagnetic force also makes possible the a…

Multivariable Calculus is an expansion of Single-Variable Calculus in that it extends single variable calculus to higher dimensions. You may find that these courses share many of the same basic concepts, and that Multivariable Calculus will simply extend your knowledge of functions to functions of several variables. The transition from single variable relationships to many variable relationships is not as simple as it may seem; you will find that multi-variable functions, in some cases, will yield counter-intuitive results. The structure of this course very much resembles the structure of Single-Variable Calculus I and II. We will begin by taking a fresh look at limits and continuity. With functions of many variables, you can approach a limit from many different directions. We will then move on to derivatives and the process by which we generalize them to higher dimensions. Finally, we will look at multiple integrals, or integration over regions of space as opposed to intervals. The goal of Mu…

This course will introduce you to the field of mechanical engineering and the relationships between physics, mathematics, communications, and sciences which inform the study, design, and manufacture of mechanical products and systems. The course is divided into four units. In the first unit, you will learn how mechanical engineering is broadly defined, what mechanical engineers do, and what technical capabilities they have. We will also review some basic principles from mathematics and physics that you will apply in any discipline of engineering. In the second unit, you will learn about the ethical considerations and technical communication skills necessary for engineering work. You will revisit these issues in more detail in several courses within the Mechanical Engineering curriculum. The third unit focuses on computational tools for engineering problems. In Unit 3 you will learn about a specific open source computational environment (Scilab) and the application of that environment to some com…

Mechanics studies how forces affect bodies in motionhow, for example, a bullet is fired from a gun or a top is set in motion by the flick of a wrist. As an engineer, you will find mechanics of vital importance to any field you choose to pursue. Whether you are designing a bridge or implementing an electrical power unit for an elevator, you will need to know how to determine which forces can be applied to a body without causing it to break, what happens when bodies collide, how an object moves when different forces are applied to it, and so on. This course will introduce you to the core concepts of mechanics that will enable you to answer these questions as you strive to design, test, and manufacture safe and reliable products. While most universities split introductory mechanics into two courses, with one devoted to statics and the other to solids, this course will introduce you to both areas. You will begin by learning about staticsobjects that are not accelerating (in other words, objects that are…

There are many different ways that you can go about solving engineering problems. One of the most important methods is energy analysis. Energy is a physical property that allows work of any kind to be done; without it, there would be no motion, no heat, and no life. You wouldn’t be able to get out of bed in the morning, but it wouldn’t matter, because there would be no sun. Without energy, our world would not exist as it does. Thermodynamics is the study of energy and its transfers though work. It is the link between heat and mechanical exertion. Once you have a solid grasp on thermodynamic concepts, you should be able to understand why certain mechanisms (such as engines and boilers) work the way they do, determine how much work they can put out, and know how to optimize these power systems. A thorough understanding of thermodynamics is crucial to any career that focuses on HVAC systems, car engines, or renewable energy technology. This course will focus on the fundamentals of thermod…

CAD, or computer-aided design, is a powerful modeling tool that technical professionals use. With CAD, architects can draw up building plans and engineers can develop component and system designs. Some CAD programs even allow users to perform stress analysis, demonstrating how well a proposed structure will fare when put to use. For example, when does a load become too big? How much weight can be put onto a bridge before it becomes structurally unsound? Using CAD, professionals can create precise engineering drawings in both 2- and 3-D, complete with dimensions and specifications, in a neat and readable format. This modeling method has taken design to a whole new level of efficiency and accuracy. We are fortunate to be engineers working in the current eraone of computers, technology, and ease of precision. Without CAD, we would have to draft (or draw up) design blueprints by hand, which can be tedious and time-consuming. With CAD, however, we can generate accurate 2-D and 3-D drawings, scale…

You may think at first that the words “fluid” and “mechanics” should not go together. However, the ways in which fluids (gases and liquids and a few other materials) respond to forces, exert forces, and move from one place to another (their mechanics) are crucially important to many aspects of our experience and our ability to build tools. Consider, for example, the following areas in which fluid mechanics play an important, if not fundamental, role: Meteorology and ocean dynamics (tsunamis, hurricanes, and tornados) Fluid flow within living systems (blood flow, lymph flow, air flow) Hydraulic machinery (jacks, pumps, lifts, steering mechanisms) Chemical processing and piping (pumps, reactors, separators, pipelines) Turbomachinery (jet engines, power plants) Aeronautical and ship machinery (airplanes, helicopters, boats and ships) In this course you will first learn about the definition of a fluid and the properties of a fluid, such as density, compressibility, and viscosity. You wil…

Dynamics is a sub-branch of the general field of study known as Mechanics. It is very closely related toand often combined withthe study of Statics, which you encountered in ME102: Mechanics I [1]. In both Statics and Dynamics, we use Newton’s 2nd Law: F = ma. In Statics, the sum of the applied forces is always zero, thus making the acceleration zero. This was very important to the structures studied in Statics. Catastrophe generally results when structures (like bridges and buildings) accelerate. Very likely you are quite pleasedeven if you do not realize it every timewhen you cross a bridge that does not accelerate while you are on it, and we have Newton’s First Law to thank for it. Newton’s First Law states that objects will continue to do what they are doing unless unbalanced forces make them do otherwise. This law includes the law equilibrium condition that the moments will also sum to zero, and that there will thus be no rotational acceleration. In Dynamics, the sum of the forces…

This self-contained course presents a sampling of the fields of Materials Engineering and Materials Science. This course is intended primarily for engineering students who are not planning to major in either Materials Engineering or Materials Science. We will focus primarily on the concerns of the materials engineerthe person interested in choosing materials to make a finished product. This selection is determined by compromises among material properties, ease of fabrication, and cost. In contrast, the materials scientist is concerned with understanding the relationships between material properties and the internal structure of a materialthat is, atomic bonding, arrangements of atoms, grain structure, and other microscopically observable features. We leave most of these associations to advanced courses, which will use more chemistry and physics than needed for this course. The course is divided into four units: Unit 1: Ways That Materials Can Fail What Can Go Wrong? Unit 2: Classes of Engineering Mate…

Heat transfer is the thermal energy in transit due to a spatial temperature difference. The topic of heat transfer has enormous applications in mechanical engineering, ranging from cooling of microelectronics to design of jet engines and operations of nuclear power plants. In this course, you will learn about what heat transfer is, what governs the rate of heat transfer, and why heat transfer is so important. You will also learn about the three major modes of heat transfer: conduction, convection, and radiation. Heat conduction is the transport of heat through a solid body, by vibrations of molecules or in the case of electrical conductors, by movement of electrons from one molecule to another. Heat convection is a process by which heat is transferred through a fluid by motion of fluid. Thermal radiation is the transport of energy between two bodies by electromagnetic waves. In addition to the three main modes of heat transfer, you will also learn about heat transfer during phase changes (boiling and conden…

Numerical methods have been used to solve mathematical expressions of engineering and scientific problems for at least 4000 years (for some historical discussion you may wish to browse the Ethnomathematics Digital Library [1] or the MacTutor History of Mathematics Archive [2] from St. Andrews University).* Such methods apply numerical approximation in order to convert continuous mathematical problems (for example, determining the mechanical stress throughout a loaded truss) into systems of discrete equations that can be solved with sufficient accuracy by machine. Numerical methods provide a way for the engineer to translate the language of mathematics and physics into information that may be used to make engineering decisions. Often, this translation is implemented so that calculations may be done by machines (computers). The types of problems that you encounter as an engineer may involve a wide variety of mathematical phenomena, and hence it will benefit you to have an equally wide range of numerical met…

This course will serve as your introduction to working in an engineering laboratory. You will learn to gather, analyze, interpret, and explain physical measurements for simple engineering systems in which only a few factors need be considered. This experience will be crucial to your success in analyzing more complicated systems in subsequent coursework and in the practice of mechanical engineering. We frequently encounter measurement systems in our everyday lives. Consider the following examples: 1. The many gauges found on the control panel of a motor vehicle indicate vehicle speed, engine coolant temperature, transmission setting, cabin temperature, engine speed, and oil pressureamongst many other measurements. 2. A routine visit to a physician often entails several measurements of varying complexityinternal temperature, blood pressure, internal appearance, heart rate, respiration rate, and tissue texture, amongst many, many more. 3. The experienced cook may use s…

Most mechanical engineering systems today involve significant amounts of electrical and electronic control systems. Effectively, most modern mechanical engineering systems are mechatronic systems. Mechatronics is the discipline that results from the synergetic application of electrical, electronic, computer, and control engineering in mechanical engineering systems. Thus, it is essential for the mechanical engineer to have a strong understanding of the composition and design of mechatronic systems, which is the goal of this course. Mechatronic systems are around us everywhere. A car contains many mechatronic systems, such as anti-lock braking systems, traction control, the engine control unit and cruise control, to name a few. A satellite dish position control unit is another example of a mechatronic system. Modern industrial automated processes would not be possible without the discipline of mechatronics, covering areas such as vehicle manufacturing, pharmaceutical industries, and food processing plants. R…

This course deals with the transfer of work, energy, and material via gases and liquids. These fluids may undergo changes in temperature, pressure, density, and chemical composition during the transfer process and may act on or be acted on by external systems. You must fully understand these processes if you are an engineer working to analyze, troubleshoot, or improve existing processes and/or innovate and design new ones. In your everyday life, you will likely encounter examples of the thermal-fluid systems we will study in this course. Consider the following scenarios: Read this recent report [1] by Gary Goettling for the Georgia Tech Alumni Association.* In it, Goettling describes a refrigeration system with no moving parts based on improvements to a patent filed by Einstein and Szilard in 1930. As an engineer, how would you go about evaluating this design for energy efficiency, safety, reliability, and manufacturing, operating, and installation costs? Have you ever wondered how the level se…

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